Z-modules

Yuichi Futa; Hiroyuki Okazaki; Yasunari Shidama

Formalized Mathematics (2012)

  • Volume: 20, Issue: 1, page 47-59
  • ISSN: 1426-2630

Abstract

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In this article, we formalize Z-module, that is a module over integer ring. Z-module is necassary for lattice problems, LLL (Lenstra-Lenstra-Lovász) base reduction algorithm and cryptographic systems with lattices [11].

How to cite

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Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. "Z-modules." Formalized Mathematics 20.1 (2012): 47-59. <http://eudml.org/doc/268180>.

@article{YuichiFuta2012,
abstract = {In this article, we formalize Z-module, that is a module over integer ring. Z-module is necassary for lattice problems, LLL (Lenstra-Lenstra-Lovász) base reduction algorithm and cryptographic systems with lattices [11].},
author = {Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama},
journal = {Formalized Mathematics},
language = {eng},
number = {1},
pages = {47-59},
title = {Z-modules},
url = {http://eudml.org/doc/268180},
volume = {20},
year = {2012},
}

TY - JOUR
AU - Yuichi Futa
AU - Hiroyuki Okazaki
AU - Yasunari Shidama
TI - Z-modules
JO - Formalized Mathematics
PY - 2012
VL - 20
IS - 1
SP - 47
EP - 59
AB - In this article, we formalize Z-module, that is a module over integer ring. Z-module is necassary for lattice problems, LLL (Lenstra-Lenstra-Lovász) base reduction algorithm and cryptographic systems with lattices [11].
LA - eng
UR - http://eudml.org/doc/268180
ER -

References

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  1. Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3):537-541, 1990. 
  2. Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990. 
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  10. Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990. 
  11. Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: A cryptographic perspective (the international series in engineering and computer science). 2002. Zbl1140.94010
  12. Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559-564, 2001. 
  13. Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990. 
  14. Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990. 
  15. Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990. 
  16. Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990. 
  17. Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990. 
  18. Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990. 
  19. Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215-222, 1990. 

Citations in EuDML Documents

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  1. Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, Quotient Module of Z-module
  2. Roland Coghetto, Groups – Additive Notation
  3. Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, Yasunari Shidama, Gaussian Integers
  4. Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, Free ℤ-module
  5. Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, Submodule of free Z-module
  6. Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, Matrix of ℤ-module1
  7. Kazuhisa Nakasho, Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, Rank of Submodule, Linear Transformations and Linearly Independent Subsets of Z-module
  8. Yuichi Futa, Yasunari Shidama, Lattice of ℤ-module
  9. Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, Yasunari Shidama, Torsion Z-module and Torsion-free Z-module
  10. Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, Torsion Part of ℤ-module

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